HAWKES PROCESSES IN FINANCE: A REVIEW WITH

SIMULATIONS

by

GRAHAM SIMON

A THESIS

Presented to the Department of Mathematics

and the Robert D. Clark Honors College in partial fulfillment of the requirements for the degree of

Bachelor of Science

June 2016

Abstract

An Abstract of the Thesis of

Graham Simon for the degree of Bachelor of Science in the Department ofMathem ·cs to be taken June 2016

Title: Hawkes Proces eview with Simulations

Hawkes processes are flexible robust models for simulating many self-exciting

features seen in empirical data. Using a Hawkes process creates clusters in modeled

data that are frequently seen in different natural environments. Some frequent areas of

use for Hawkes processes include the study of earthquakes, neural networks, social

media sharing, and financial trading data. This work builds an accessible framework for

the undergraduate study of Hawkes processes through building step-by-step from point

processes to Poisson processes and eventually Hawkes models. A literature review of

current research and utilizations for Hawkes processes is then done to demonstrate some

of the dramatic growth seen in this field of research. Point clusters, kernel estimation,

parameter estimation, and algorithms for implementation are also discussed with simple

simulations performed in Excel.

H

iii

Acknowledgements

First, I would like to thank Professor Chris Sinclair for all his help and guidance

during the thesis process. Not only is Chris a wonderful guide in the labyrinth of

mathematics, but he also provided the prospective, wisdom, and encouragement to

make this research process a truly multifaceted learning experience. Second, I would

like to thank Hayden Harker, Eric Merchant, and Aaron Montgomery for their

wonderful instruction and advice during my time at the University of Oregon. Finally, I

would like to thank my grandparents, parents, and wife for their unwavering love and

support in this and all my pursuits.

iv

Table of Contents

Introduction 1

Undergraduate Research of Hawkes Processes 3

Theoretical Background 7 Point Processes 7 Point Process Models 8

Overview of Hawkes Processes 14

Hawkes Process 14 Initial Kernel Development & Relationship with Autoregressive Models 16 Different Kernels 18 Beginning Analysis of Point Clusters 20 Data Simulation 22 Estimation of the Kernel 23

Literature Review 27 Other Interesting Areas of Self-Exciting Point Process Research 30

Algorithms for Simulation and Estimation 32 Poisson Processes 32 Univariate Hawkes Process with Exponentially Decaying Intensity 36 Multivariate Hawkes Process with Exponentially Decaying Intensity 42 Estimation of a Power Kernel 44 Implemented Processes 46

Future Work 49

Improve Simulation Implementations 49

Bibliography 50

v

List of Figures

Figure 1: Graph of Exponential Distribution with λ = 5 9

Figure 2: Graph of Exponential Density Function with λ = 5 9

Figure 3: Graph of Poisson distribution with λ = 5 10

Figure 4: Graph of Poisson Densifabty Function with λ = 5 11

Figure 5: Kernel estimate showing individual kernel. Window width of 0.4 (Silverman, 1986). 26

Figure 6: Graph of −1𝜆

ln 𝑈 for λ = 56T 33

Figure 7: Example of a Generic Exponential Intensity Decay 38

Figure 8: Homogenous Poisson Process with λ = 4 and T = 50 46

Figure 9: Non-homogenous Poisson Process: λ = 7.5, T = 7.5, & λ(t) = 𝑛𝜏𝑒𝑒𝑒 �− 𝑡

𝜏� 46

Figure 10: Intensity of Univariate Hawkes Process: a = 0.45, δ = 0.25, λ0 = 0.9, & Intensity Jumps of Size 0.40 47

Figure 11: Histogram of Number of Events per Unit Time for Univariate Hawkes Process: a = 0.45, δ = 0.25, λ0 = 0.9, & Intensity Jumps of Size 0.40 47

Introduction

Properly functioning capital markets (e.g. stock, bond, and real estate markets)

not only provide efficient capital allocation, but also the necessary gains for the funding

of pension funds, retirement accounts, and university endowments. Ideally these

important institutions undertake their fiduciary duty free from speculation and

undertake an investment operation “which upon thorough analysis, promises safety of

principal and a satisfactory return” (Graham and Dodd, 1962). By undertaking this

process investment managers seek not only objectively sound returns, but also risk

adjusted returns above the average market participant. However, the existence of such a

process has been one of the most debated financial subjects of the last century. Most

academics have now concluded sustained outperformance is impossible and profess

belief in the efficient market hypothesis. Under the efficient market hypothesis, all

public and nonpublic information is continuously priced into financial assets so any

performance above the average must be short term luck. Thus after a period of

outperformance a fund will likely enter a period of underperformance pushing the

individual average down to, or below (after fees), the larger industry average.

Against this academic backdrop, speculators have spent centuries attempting to

profit from short term fluctuations in the value of stocks, bonds, commodities, and other

investment vehicles. George Soros offered an explanation for this phenomenon in his

book The Alchemy of Finance through the idea of market reflexivity. Market reflexivity

presents the idea that financial markets combine an interplay of fundamental exogenous

events (breaking news, new financial results being release, and other economic

fundamentals) with market created endogenous events (price changes from previous

2

trades or other purely market based forces). Reflexivity carries especially high weight

today since different historical bubbles have spawned legions of day traders and other

speculators, but modern trading occurs primarily through computers. Computerized

trading allows for much larger blocks of stocks or bonds to be rapidly moved for minute

gains unachievable by slower human traders. This means rigorous analysis may now

examine trading data in time intervals as small as a microsecond, and focus upon the

system itself as much as the economic fundamentals affecting the financial asset.

Unlike stocks and bonds, which are tied to the future cash flow generations of

companies or governments, commodities (e.g. oil, natural gas, corn, and cocoa) are

exchange traded assets with no value beyond their eventual use. In commodity financial

markets future contracts are bought and sold that give the owner the right to buy or sell

a certain commodity at a predetermined price sometime in the future. For example one

of the most common oil trades is the price currently agreed to be paid upon delivery of a

barrel of West Texas Intermediate (WTI) oil in Cushing, Oklahoma one month in the

future. Companies use commodity contracts to help set and plan for the prices received

and paid for their good and inputs in the future. However, companies do not act alone in

this arena as speculators thrive, and die, guessing upon potential changes in commodity

prices. Overall, based on the uniform character of commodities these assets tend to

experience greater endogenous effects than stocks and bonds and trade in ways heavily

influenced by the market itself.

When building an initial framework for understanding short term trading in this

area, prior econometricians realized they needed a model with a “time series of

irregularly spaced points that show a clustering behavior” (Fonseca and Zaatour, 2016).

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During the financial trading day the clustering of orders appears most obviously at the

beginning and ending of the day, but orders between these heavy times also tend to

clump together. Alan Hawkes first popularized a quickly understood process fulfilling

this criteria in 1971. Now known as a Hawkes process this model created a self-exciting

process (i.e. one event increases the probability another event will follow shortly) with

exponential (rapid) time decay mimicking the clustering of neuron firing, earthquakes,

and financial trading data. As high frequency trading began representing a larger and

larger portion of total trading volume during the 1990’s and 2000’s the study of Hawkes

processes also became a more active area of research.

This paper will first review the mathematical creation and improvement of

Hawkes models before exploring its financial capabilities and end with simulations of

different point processes. This thesis should serve as a roadmap to the necessary

material to learn about Hawkes processes, its flexibility, and potential utilization in

many different fields. Unfortunately while this model seems well taught and utilized at

the graduate level, its recent development means little writing exists at an undergraduate

level. With time this will most likely change due to the ever expanding amount of

available electronic data in modern society and the need for rapid and precise responses

to high intensity events. Finally, simulations are generated for homogenous and

nonhomogeneous Poisson processes and a univariate Hawkes process.

Undergraduate Research of Hawkes Processes

This thesis is designed to be accessible to undergraduate students who have

previously taken courses in probability or statistics and wish to learn about the unique

properties of Hawkes processes. However, a couple notes should be made about

4

studying these processes and the approach undertaken within this thesis. First,

attempting to implement Hawkes processes, like most advanced models, requires a fair

amount of coding. In order to fully research these processes, it is therefore necessary to

spend a lot of time gaining familiarity with how to implement probability code, in

addition to learning the rigorous underpinnings of the Hawkes process. Possessing both

these skills then allows researcher the opportunity to quickly adapt or tune a model to

best fit empirical data. Given the range of skills necessary for this it is therefore no

surprise that a couple quantitative finance research departments produce an outsized

proportion of current research on this topic.

This thesis focuses on the knowledge necessary to understand and

discuss Hawkes processes, but utilizes more simplistic models for simulations.

Additionally, while there are certainly many interesting avenues for applied research

with Hawkes processes, not enough progress was made to construct models from

empirical data. Not reaching this point was certainly a disappointment, but it also serves

as a highly instructive learning moment on the difficulties of research. Unfortunately

progress does not occur linearly, and gauging the full scope of these projects cannot be

seen until they are well underway. Eventually research for this thesis converged upon

understanding Hawkes process, their properties and attributes, and a couple algorithmic

ways of simulating point processes.

One of the biggest areas for expansion of this thesis, if research was to be

continued, would be to develop a better understanding of maximum likelihood

estimation and other forms of parameter estimation. Since methods like maximum

likelihood estimation provide the numerical values for empirical modeling, a more

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robust understanding of these techniques would be paramount to progressing further in

the subject matter. Greater time spent coding and building a firm understanding of

statistical software would also be critical to adding new results. Assuming this

knowledge was acquired, accumulating the necessary financial data to create empirical

models would be the next difficult step.

There are a couple different financial services firms that maintain data detailed

enough for the full benefit of modeling with Hawkes processes, but access to these data

sets costs in excess of $20,000 and cannot be replicated from free sources. Methods,

like a Brownian bridge, exist to distribute accumulated discrete points across a range in

order to create smaller breaks in the data set, but these then alter the underlying

structure of the data. High costs associated with financial data therefore offer another

reason why authors from the same departments and centers possess the best data

available for modeling Hawkes processes.

Finally, while research in this area seems to have done a good job of better

describing some attributes of financial markets, even the best model fails to present a

compelling opportunity for profits. While it is quite possible that researchers with

profitable ideas quietly implemented them, and reaped profits themselves, the available

literature focuses more on the complexity of modern financial markets. In particular

researchers demonstrate the rapidity of change within financial markets and how little

short term financial trading resembles the trading world of only a decade ago. Thus,

progression from new research requires simultaneous progress in both the applied and

theoretical aspects.

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New theoretical ideas must be created, implemented, tested against empirical

data, and then revised in order to make meaningful improvement. Attacking these

problems is therefore quite difficult at an undergraduate level, but the Hawkes process

is flexible enough that benefits from using it may eventually be found in a multitude of

yet unknown areas.

7

Theoretical Background

Point Processes

In order to analyze the effects of many scientific and financial processes it is

necessary to begin counting the frequency of events over time. Mathematically, keeping

track of when random events occur during a known time window is a point process.

Each point then represents a “time and/or location of an event, such as a lightning

strike,” earthquake, or stock trade (Schoenberg, 2016). In general a point process, N, is

defined as a random increasing step function on a “metric space S taking values in the

non-negative integers” (Schoenberg, 2016). This simply means that N is a function

which represents an integer count between 0 and infinity (inclusively) for the number of

points filling in a subset A of S. Even more simply N is just a function counting the

number of events during any time window. While an infinite number of points may

appear in a given subset, most point processes building upon real world data remain

focused upon areas where N may contain “only finitely many points on any bounded

subset of S” (Schoenberg, 2016). Focusing on situations with a finite number of events

then allows for applied analysis to reach meaningful conclusions.

Restricting this general definition to the needs of this paper we can consider a

temporal point process (all events occur between times 0 and T). For a temporal point

process N is then simply an ordered list {t1, t2, ..., tn} of event times. Alternatively the

list may be thought of as inter-event times ui = ti - ti-1 and provide the list of gaps

between events {u1, u2, ..., un}, taking t0 = 0. In order to get the specific count of events

at any positive time t < T we may then use the notation N(t) to reference the number of

8

points occurring at or before time t. The process N(t) will then be non-decreasing (since

an event cannot unhappen), right-continuous, take only non-negative integer values

(there cannot be negative events), and have left jump discontinuities at each event time

tj (when a new event pushes the function value up one). Thus, a temporal point process

could alternatively be defined “as any non-decreasing, right-continuous Z+-valued

process” (Schoenberg, 2016). Defining point processes this way then fulfills all the

criteria necessary and immediately denotes the flexibility of allowing any such function.

A point process is called simple if, with probability one, all its points ti occur at distinct

times and orderly if for any time t, 𝑙𝑙𝑙𝛥𝑡 ➝ 0

𝑃{𝑁(𝑡,𝑡 + 𝛥𝑡] > 1}𝛥𝑡

= 0.

Beyond simply time values, a point process can also contain additional variables

to make it a marked point process (i.e. a function with multiple input variables). A

financial example of a marked point process would be a set which not only contains the

times of different market trades, but also the sizes of the trades, whether it moved the

quoted price, or who made the trade. Marked point processes are also then very similar

to time series data. In principle the different realizations of a marked point process

could be viewed as the dataset of a time series and vice versa. However, a time series

dataset and a marked point process differ by allowing an event to take any time in a

continuum, whereas as time intervals are deterministic for time series data.

Point Process Models

Before defining the most common point process models recall that a random

variable T is said to have an exponential distribution with rate λ > 0, or T =

exponential(λ), if:

9

P(T < t) = 1 - e-λt for all t > 0.

Figure 1: Graph of Exponential Distribution with λ = 5

This will then produce a density function fT(t) equal to:

𝑓𝑇(𝑡) = �λ𝑒−λ𝑡 𝑡 ≥ 0

0 𝑡 < 0.

Figure 2: Graph of Exponential Density Function with λ = 5

Using these definitions we may then define one of the most utilized point process

models, and the most important one for this paper, the Poisson process (Durrett, 1999).

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Definition: Let 𝜏1, 𝜏2, ... be independent exponential(λ) random variables. Let Tn

= 𝜏1 + 𝜏2 + ... + 𝜏n for n > 1, T0 = 0, and define N(s) = max{n : Tn < s}.

Homogenous Poisson processes (i.e. those with a constant value for λ) N(s) are then

distributed with mean λs and variance λs.

Figure 3: Graph of Poisson distribution with λ = 5

One of the nice attributes of Poisson processes are that they have independent

increments. This means that a Poisson process N(t) is such that if t0 < t1 < .... < tn, then

N(t1) - N(t0), N(t2) - N(t1), ..., N(tn) - N(tn-1) are independent random variables. Using

this language a Poisson process may also be defined as (Durrett, 1999):

Theorem: If {N(s), s > 0} is a Poisson process then: (i) N(0) = 0 (ii) N(t + s) - N(s) = Poisson(λt) (iii) N(t) has independent increments

Conversely, if (i), (ii), and (iii) hold, then {N(s), s > 0} is a Poisson process.

00.10.20.30.40.50.60.70.80.9

1

0 1 2 3 4 5 6 7 8 9 10 11 12

t

N(t)

11

Figure 4: Graph of Poisson Density Function with λ = 5

However, these theoretical models have a flaw in that most financial events do not

occur with uniformly over time. Therefore, a modified Poisson process with a varying

rate for λ can be much more accurate, and is called a Non-homogenous Poisson process.

Such a process is defined as:

{N(s), s > 0} is a Poisson process with rate λ(r) if: (i) N(0) = 0 (ii) N(t) has independent increments (iii) N(t) - N(s) is Poisson with mean ∫ 𝜆(𝑟)𝑡

𝑠 𝑑𝑟.

The function λ(t), represents the infinitesimal rate at which events are expected

to occur around a particular time t. One way of thinking of this would be to consider it a

hurdle rate for a new randomly generated value to have to clear in order for a point

there to be included (this will actually be part of the method of modeling non-

homogenous processes later). This rate is known as the conditional intensity of the non-

homogenous Poisson process and is based on the current time in the point process.

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Estimating λ(t) can be done both parametrically and nonparametrically (i.e. with

external constraints or without external constraints), to allow λ(t) to vary properly will t.

For a temporal point process originating at time 0 the compensator (the average

intensity), A(t), may be defined as the integral of the conditional intensity from time 0 to

time t. An equivalent definition would be the compensator is the unique non-negative

non-decreasing predictable process A(t) such that N[0,t) - A(t) is a martingale (Note:

N[s,t) = N(t) – N(s)).

A renewal process is a point process where the inter-event times {u1, u2, ..., un}

are independent but not necessarily exponential random variables. Density functions

governing each inter-event time are thus known as renewal density functions. Such

models describe situations in which the probability of an event occurring depends only

on the time since the most recent event (e.g. in fire hazard analysis such a model is

consistent with wood fuel loading followed by complete fuel depletion in the event of a

fire). Overall these characteristics provide the ability to generate a point process with

varying degrees of intensity based on the history of events up to any time t. However,

these processes can still be improved further by adding factors that create the clustering

behavior seen in many market situations.

Setting the stage for Hawkes processes comes the idea of self-exciting and self-

correcting point processes. A point process N is self-exciting if cov{N(s,t), N(t, u)} > 0

for s < t < u and is self-correcting if the covariance is negative. This means “the

occurrence of points in a self-exciting point process causes other points to be more

likely to occur, whereas in a self-correcting process, the points have an inhibitory

effect” (most simply self-exciting processes clump together in bunches because one

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event increases the chances of another happening while self-correction processes spread

out event times through the opposite process) (Schoenberg, 2016). Connecting this to

the previous discussion of homogenous and non-homogenous Poisson processes this

means the intensity of self-exciting and self-correcting processes are dependent on

previous events in the same way a non-homogenous Poisson process has varying

intensity with time. While a Poisson process is neither self-exciting nor self-correcting

by definition, λ(t) may be modified to produce results similar to a self-exciting or self-

correcting process. However, it is only through the use of Hawkes processes that point

processes most naturally cluster and mirror empirically observed phenomena.

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Overview of Hawkes Processes

Hawkes Process

A Hawkes process is a point process with a response function (or kernel) ϕ(t - ti)

which reflects the influence of past events on the conditional intensity. The beauty of

this model is that it is more general than the Poisson process and has greater potential to

explain some of the phenomena seen in financial markets. Specifically, Daley and Vere-

Jones claim the Hawkes process, “comes closest to fulfilling, for point processes, the

kind of role that the autoregressive model plays for conventional time series” (Daley,

2003). Given a counting process N(t) = max{i : ti < t} and filtration Ft- = {t1, ..., ti: ∀i <

N(t)}, representing the information about the process up to time t, a linear continuous

Hawkes process may be defined as a point process {ti}i ∈ Z+ with conditional intensity

given by:

𝜆(𝑡 | 𝐹𝑡−) = 𝜇(𝑡) + � 𝜙(𝑡 − 𝑠)𝑡

−∞𝑑𝑑(𝑠).

The conditional intensity is defined as:

𝜆(𝑡 | 𝐹𝑡−) = 𝑙𝑙𝑙ℎ →0

𝐸[𝑑(𝑡 + ℎ) − 𝑑(𝑡) | 𝐹𝑡−]ℎ

. Intuitively “the conditional intensity is an infinitesimal expected rate at which the

events occur around time t given the history of the process N before t” (Morzywolek,

2015).

Within this model the two most important moving pieces are then μ(t) and the

kernel function ϕ. Generally μ(t) is seen as the background intensity responsible for

accounting for the arrival of exogenous (external) events while the kernel function ϕ,

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satisfying causality condition ϕ(t) = 0 for t < 0, determines the correlation properties of

the process. (Note: Throughout the rest of this paper exogenous events will frequently

be referred to as immigrant events since these events were caused by external factors,

and events generated by ϕ will be referred to as descendants or children since these are

system created events “descended” from the initial immigrant.) Then the branching

ratio, n, for the process may be defined as

𝑛 = � 𝜙(𝑡)∞

0𝑑𝑡.

The differential of the counting process may then be rewritten as a sum of delta

functions

𝑑𝑑(𝑡) = � 𝛿(𝑡 − 𝑡𝑖)𝑡𝑡 < 𝑡

.

To form the discrete time form of the conditional intensity

𝜆(𝑡 | 𝐹𝑡−) = 𝜇(𝑡) + 𝑛 � ℎ(𝑡 − 𝑡𝑖)𝑡𝑖 < 𝑡

where h(t) = ϕ(t)/n is called a bare kernel. Note that the bare kernel is a probability

density function and the Hawkes process is stationary for n < 1. Stationarity implies if

X1 and X2 are independent copies of a random variable with a,b > 0 being constants

then aX1 + bX2 has the same distribution as cX + d for some positive constants c,d.

Assuming stationarity and taking μ(t) to be constant, which will simply be

denoted µ from now on, the average total intensity Λ may be calculated as

Λ = 𝐸[𝜆(𝑡 | 𝐹𝑡−)] = 𝐸[ 𝜇(𝑡) + � 𝜙(𝑡 − 𝑠) 𝑡

−∞𝑑𝑑(𝑠)] = 𝜇 + Λ� 𝜙(𝜏)

∞

0𝑑(𝜏)

which implies

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Λ =𝜇

1 − 𝑛.

Note that if n > 1 this formula implies that Λ→ ∞ exponentially quickly, and hence the

counting process N(t) eventually explodes. Hence understanding the branching ratio is

critical to properly analyzing a Hawkes process. Essentially the branching ratio

represents the average number of first-generation daughters (market created events) of a

single mother (actual company or economic event). If n = 0 then the model collapses

back down to a non-homogenous Poisson process since there will be no further events

triggered by the initial immigrants. Therefore, the Hawkes process may be viewed as a

generalization of the Poisson process that depends on both the time and history of a

process. Further, the critical case n = 1 separates the model into subcritical (n < 1) and

supercritical (n > 1) states. If n > 1 for a sustained amount of time the modeled intensity

may explode beyond applied analysis so most study focuses upon subcritical cases.

Finally, “whenever the intensity µ is a constant and the process is in the subcritical (n <

1) or in the critical (n = 1) regime the branching ratio can be used as a measure of the

proportion of events that are generated inside the model (by the presence of the

exponential kernel, i.e. endogenously generated events) to all events” (Lorenzen, 2012).

Initial Kernel Development & Relationship with Autoregressive Models

Early development of the Hawkes process focused on the exponential kernel ϕ(t

- ti) = 𝛼𝑒−𝛽(𝑡−𝑡𝑖)which leads to the conditional intensity

𝜆𝑡(𝑡) = µ(𝑡) + � 𝛼𝑒−𝛽(𝑡 − 𝑡𝑖)

𝑡𝑖 < 𝑡.

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To begin seeing how this process resembles an autoregressive model consider the

intensity at some past specified time ti. Then the intensity will be

𝜆(𝑡𝑖) − µ(𝑡𝑖) = � 𝛼𝑒−𝛽(𝑡𝑖 − 𝑡𝑘)

𝑡𝑘 < 𝑡𝑖

where tk represents all events that occurred before ti. Next if we multiply both sides of

the previous equation by exp[-β (t - ti)] we have

[𝜆(𝑡𝑖) − µ(𝑡𝑖)]𝑒−𝛽(𝑡 − 𝑡𝑖) = � 𝛼𝑒−𝛽(𝑡 − 𝑡𝑘)

𝑡𝑘 < 𝑡𝑖.

Now the response function can be decomposed into

� 𝛼𝑒−𝛽(𝑡 − 𝑡𝑖)

𝑡𝑖 < 𝑡 = � 𝛼𝑒−𝛽(𝑡 − 𝑡𝑘)

𝑡𝑘 < 𝑡𝑖+ � 𝛼𝑒−𝛽(𝑡 − 𝑡𝑘)

𝑡𝑘 > 𝑡𝑖 < 𝑡.

Combining the last two equations we can write λ(t) - µ(t) as

𝜆(𝑡) − µ(𝑡) = [𝜆(𝑡𝑖) − µ(𝑡𝑖)]𝑒−𝛽(𝑡 − 𝑡𝑖) + � 𝛼𝑒−𝛽(𝑡 − 𝑡𝑘)

𝑡𝑘 > 𝑡𝑖.

This then deeply resembles the continuous time form of an autoregressive model

Xt - μ = e-β(t - s) (Xs - μ) + sum of innovations

“where the term [λ(ti) - µ(ti)]𝑒−𝛽(𝑡 − 𝑡𝑖)is the autoregressive term and the term

∑ 𝛼𝑒−𝛽(𝑡 − 𝑡𝑘)𝑡𝑘 > 𝑡𝑖 represents the sum of the innovations in the AR process” (Lorenzen,

2012).

For the exponential kernel the unconditional intensity Λ for a trading day may

be calculated as

𝐸(𝜆) = 𝐸(𝜇) + 𝐸 �� 𝛼𝑒−𝛽(𝑡−𝑠)𝑡

−∞𝑑𝑑(𝑠)�.

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Then assuming stationarity as above we get the expected intensity as

𝐸(𝜆) = 𝜇

1 − 𝛼𝛽.

Hence for the exponential kernel the characteristics and behavior of the point process

are determined by the ratio of 𝛼𝛽

since

𝑛 = � 𝛼𝑒−𝛽𝑡∞

0𝑑𝑡 =

𝛼𝛽

.

Different Kernels

In many applied settings simply observing the distribution of events provides

enough evidence to decide what “the kernel should look like or what properties it

should have” (Morzywolek, 2015). However, there are still a couple different options

for the kernel in a Hawkes process. First, the power law kernel is defined as

ϕ𝑝𝑝𝑝(𝑡) =𝑛𝑛𝑐𝜃

(𝑡 + 𝑐)1+𝜃 𝜒(𝑡)

with 𝜒(𝑡)representing the unit step function (i.e 𝜒(𝑡)is 0 if t < 0 and 1 if t > 0). The unit

step function guarantees the causality and is often used in geophysical applications.

Second, the exponential kernel can be written in a slightly modified form from ϕ(t - ti)

= 𝛼𝑒−𝛽(𝑡−𝑡𝑖)and be defined as

𝜙𝑒𝑒𝑝(𝑡) =𝑛𝜏𝑒𝑒𝑒 �−

𝑡𝜏� 𝜒(𝑡).

19

Exponential kernels provide a much faster decay in the probability distribution than

power kernels and are generally used in short-memory processes. Finally, a couple

modified kernels such as the cut-off kernel (Kagan and Knopoff, 1981)

𝜙𝑐𝑐𝑡(𝑡) =𝑛𝑛𝜏0𝑡1+𝜖

𝜒(𝑡 − 𝜏0)

and the double exponential kernel (Rambaldi, Pennesi, and Lillo, 2014).

ϕ𝑑𝑒(𝑡) = �𝛼1exp �−𝑡𝜏1� + 𝛼2exp �−

𝑡𝜏2�� 𝜒(𝑡)

have been suggested to help bridge the gap between the two most common kernels.

Specifically for financial processes there continues to be debate about which

kernel best represents the dependencies of price changes in financial assets, but the

exponential kernel is more frequently utilized. Despite this frequency the power kernel

and its longer term price correlation are still advocated by many (Hardiman, Bercot, and

Bouchaud, 2013) (Bacry, Dayri, and Muzy, 2012). This contrasts with the general

consensus which maintains previous price movements have limited impact and

therefore exponential kernels make more sense (Filimonov and Sornette, 2012)

(Filimonov and Sornette, 2015). Arguments for the exponential kernel also include the

Markov property frequently assumed for financial assets. Financial models often

assume the Markov property, that previous trading data does not affect the future

probabilities for an asset, which lends credence to the short-memory process and kernel.

Finally, the nature of the financial data being studied also affects the selected kernel.

Data sets with frequent sampling, such as high frequency trading, tend to use

exponential kernels more often than longer time horizon studies that use power kernels.

20

This is because things like stock market trades depend far less on long ago trades than

corporate bond default rate trends that can take months or years to play out.

Beginning Analysis of Point Clusters

As alluded to above, the beauty of a Hawkes point process model is the point

clusters generated by these models. Clusters emerge because an initial immigrant event,

from an external news or economic event, causes the probability of a subsequent event

occurring to increase. This increase in the probability of another event occurring makes

the Hawkes process self-exciting and capable of modeling the clustering of data often

seen in empirical data. While this might not initially seem like a groundbreaking

discovery it allows for the creation of models much more in tune with reality than many

deterministic models. In everything from the dynamics of views of YouTube views

(MacKinlay, 2015) and Twitter retweets (Zhao, Erdogdu, He, Rajaraman, and

Leskovec, 2015) to analyzing earthquake data a wide variety of point processes occur in

clusters. Financial trading data also clusters around big news events, certain times of

day, and other catalysts.

Looking beyond the surface level clustering of a point process one of the next

questions is how overlapping or separated individual clusters are. In order to view the

separation, or lack thereof, in clusters a couple quick definitions are necessary. First, a

renormalized kernel describes the responses of points within a cluster to the initial

immigrant point. This is in contrast with the previously discussed bare kernels which “is

nothing else than the probability of a mother event triggering a daughter event”

(Morzywolek, 2015). Hence using a renormalized kernel it is possible to talk about the

overlapping nature of clusters within the point process.

21

Bare kernels, h(t), and renormalized kernels, R(t), are related by the equation

ℎ(𝑡) = 𝑅(𝑡) − 𝑛� ℎ(𝑡 − 𝑠)𝑡

0𝑅(𝑠)𝑑𝑠.

For the exponential kernel the renormalized kernel is

𝑅(𝑡) = 1𝜏

𝑒𝑒𝑒 �−𝑡 (1 − 𝑛)

𝜏�.

This example shows that the renormalized kernel is not a probability density function

(PDF), but in general

� 𝑅(𝑡)∞

0𝑑𝑡 =

11 − 𝑛

.

Using this information it is then possible to calculate the average distance between

immigrants and the average length of a cluster. Assuming as above that immigrants are

generated from a homogenous Poisson process with intensity µ the average distance

between points is then 1𝜇

, but the occurrence of descendants is defined by a renormalized

kernel like 1𝜏

exp �− 𝑡 (1−𝑛)𝜏

�.

This means the average length of the cluster is

� 𝑡 (1 − 𝑛)∞

0𝑅(𝑡) 𝑑𝑡 = �

(1 − 𝑛)𝜏

𝑡 exp �−(1 − 𝑛)

𝜏𝑡�

∞

0 𝑑𝑡 =

𝜏1 − 𝑛

where the (1 - n) factor accounts for the renormalized kernel not being a PDF. With this

information 𝜅 will be defined as the ratio of the average length of the cluster and the

average distance between immigrants. If 𝜅 < 1 it follows that average cluster lengths are

shorter than the average distance between immigrants and therefore the point clusters

are well-separated. However, if 𝜅 > 1 then cluster lengths are much longer than the

average distance between immigrants and most clusters overlap. Finishing the example

22

of a renormalized exponential kernel the value of 𝜅 for this renormalized kernel is

simply

𝜅 =𝜏/(1 − 𝑛)

1/𝜇=

𝜇𝜏1 − 𝑛

.

Data Simulation

Currently there are two commons ways to generate a simulated Hawkes process.

The first, the thinning method, was initially proposed by Lewis and Shedler in 1978 and

was used to simulate inhomogeneous Poisson processes. Ogata then applied the

thinning method to the Hawkes process in 1981 and instituted it as one of the most

common ways of simulating a Hawkes process. A thinning process works by first

generating data points t1, ..., tN from a homogenous Poisson process with intensity λmaj

being majorant to the conditional intensity λ(t) of the Hawkes process being generated,

i.e. λmaj > λ(t), ∀t. The majorant intensity then acts as a filter for the acceptance-

rejection method. Using randomly generated values for a given point ti points are then

accepted with probability pi given by

𝑒𝑖 =𝜆(𝑡𝑖)𝜆𝑚𝑚𝑚

and otherwise removed from the sample.

Unfortunately there are two drawbacks to using the thinning method of Hawkes

process generation. First, this process runs in O(N2) time which makes it rather

inefficient when dealing with large near-critical Hawkes processes. Second, using this

method also ignores the branching structure of the process and generates events created

by both exogenous and endogenous factors simultaneously. Simulating events in this

23

manner makes it impossible to tell whether event is an exogenous immigrant or an

endogenous child of a previous immigrant event. In order to deal with these

shortcomings, more modern studies (Møller and Rasmussen, 2005) (Møller and

Rasmussen, 2006) have suggested simulating all clusters in parallel generation by

generation.

Parallel generation begins by simulating all the immigrant events from a

homogeneous Poisson process with intensity λ(t) equal to the background intensity of

the Hawkes process μ. Next the first generation of descendants, which begins the

cluster, is modeled for each immigrant event ti. This is done by utilizing a

nonhomogeneous Poisson process with the intensity λi(t) = ϕ(t - ti). Similarly once the

kth generation has been constructed the points of the (k+1)th generation are produced by

a nonhomogeneous Poisson process with intensity λk,i(t) = ϕ(t - tk,i) (where tk,i is the ith

point in the kth generation of a cluster). Finally, this process is then repeated in parallel

until there are no more offspring. Utilizing this process then allows the numerical

complexity to decline to O(μTK) where T is the size of the simulation window and K is

the number of generations modeled. Additionally, this process allows for the

reconstruction of the entire branching structure in order to determine the attributes of

point clusters.

Estimation of the Kernel

Estimation of the kernel for a Hawkes process is of critical importance due to

the large impact it has upon the shape and characteristics of a model. Since different

kernels have their intensity decay at different rates, and different parameters will

enhance or diminish these differences, it’s important to estimate an accurate kernel.

24

Once a kernel has been estimated it is then used as the determining factor for the

intensity of the Hawkes process at different times and therefore drives the attributes of

the model. The values for a kernel also determine whether modeled point clusters are

disjoint or overlapping. When estimating point clusters that are disjoint it might be

possible to simply eye ball a model with reasonable accuracy, but if knowing the

generations of an initial immigrant or the independence of clusters is important to a

researcher they must estimate a kernel that properly reflects the underlying system.

Proper parameter estimation also ensures that useful measurements like the average

total intensity and branching ratio are calculated accurately. Therefore, without the

proper estimation of a kernel it is difficult to model characteristics that are in real

conformity with the studied point process.

When seeking to numerically estimate the parameters for a kernel the most

popular method is Maximum Likelihood Estimation (Ogata, 1998). For the Hawkes

process the log-likelihood function associated with it is

log 𝐿(𝑡1, . . . , 𝑡𝑁) = � 𝑙𝑙𝑙 𝜆(𝑡𝑖 | 𝐹𝑡𝑖−)𝑁

𝑖=1 −� 𝜆(𝑡 | 𝐹𝑡−)

𝑇

0𝑑𝑡

where t1, ..., tN ∊ (0, T] are the observed events. Using this function the parameters of

the Hawkes model can then be calculated numerically by maximizing the log-likelihood

function in the case of both the exponential and power law kernel. One of the more

easily understood methods of estimating these values comes from Kernel Density

Estimation. This process attempts to use time sections of varying size (described by

their window width) to construct a smooth approximation of a kernel of a Hawkes

process.

25

First, a potential kernel K must satisfy the condition that

� 𝐾(𝑒) ∞

−∞𝑑𝑒 = 1.

Most of the time K will be a symmetric probability density function, but this does not

always hold. Two of the most widely used kernels are the boxcar kernel

𝐾(𝑒) = 12

1[−1,1](𝑒)

and the Gaussian kernel

𝐾(𝑒) = 1

√2𝜋𝑒𝑒𝑒 �−

𝑒2

2�.

Next, the kernel estimator, 𝑓,� formed from l real observations X1, ..., Xl with kernel K

will be

𝑓(𝑒) = 1ℎ𝑙� 𝐾�

𝑒 − 𝑋𝑖ℎ

�𝑙

𝑖 = 1

where h is the window width, length of a sub-sections, within the total time span. The

kernel function K then determines the shape of event “bumps” while the window width

h determines their width. Figure 1 demonstrates an example of this where individual

bumps, l-1h-1K{(x-Xi)/h}, are shown as well as the estimate of 𝑓given by adding them

up.

26

Figure 5: Kernel estimate showing individual kernel. Window width of 0.4 (Silverman,

1986).

Finally, the approximation of the kernel of the Hawkes process is of the form

𝜙� = 1

2𝛿𝛿 � � 𝐾�

𝑡 − |𝑡𝑖 − 𝑡𝑚|ℎ

� .𝑁

𝑚 = 1

𝑁

𝑖 = 1

This method of kernel approximation provides for a couple nice properties.

First, if the kernel is everywhere non-negative and satisfies (1) then the kernel is a

probability density function and therefore 𝑓will also be a probability density.

Additionally, 𝑓 inherits all the continuity and differentiability properties of the

underlying kernel. One example of this would be the fact that the Gaussian kernel

immediately implies 𝑓will be a smooth curve with derivatives of all orders. However,

one drawback to this method can be seen when estimating kernels with long-tailed

distributions. Since the window width is fixed across the entire sample, spurious noise

in the tails can cause spikes within 𝑓.� Rectifying this flaw then requires the estimate

being smoothed further, but this then begins to dilute the accuracy of the main bulge of

the distribution, i.e. the main spike will be decreased at the expense of minimizing the

spike in the tail of the distribution.

27

Literature Review

Beyond the works referenced in the earlier introduction and theoretical overview

a few more deserve especial attention. First, while the Hawkes model was initially

proposed in 1970’s it took almost 30 years for this point process to gain traction in the

financial community. Bowsher (2007) and Hewlett (2006) were two of the first to

utilize Hawkes models when they studied mid-quote (price between the bid and ask

prices) changes and order flow in the FX (foreign currency exchange) market

respectively. Generalized Hawkes processes, described in terms of vector conditional

intensities, are utilized by Bowsher to demonstrate a two-way interaction between

trades and changes in mid-prices within General Motors Corporation (GM) stock during

a 40 day study window from 2000. The first direction of influence is that the occurrence

of a trade increases the intensity of mid-price changes, and the logical second influence

is that mid-price changes do in fact increase trade intensity (i.e. trading volume and

mid-price movements are positively correlated).

At the time available databases used timestamps with only one second of

precision. Combined with the budding growth of high frequency trading this meant

several orders were often combined into one second despite happening at materially

different times. In order to combat this issue Bowsher set a precedent by adding a

uniform random component that distinguished between equal timestamps (most current

published studies now have access to data sets with precision greater than one second).

For the GM data set studied only 0.26% of all trades shared a timestamp, but by 2010 in

one trading day in February Yahoo’s stock had ~30% of all trades share a second with

at least one other trade. Such a rapid increase in the frequency of executed trades then

28

helps the stage for later study of the power high frequency trading has upon financial

asset prices. Very close to the publishing of Bowsher’s work Hewlett used a bivariate

Hawkes process to predict future FX trading intensity conditional on recent trades.

Hewlett utilized a bivariate Hawkes process because in that market liquidity takers

(brokerage firm or other market maker) that need to fill a large order are faced with a

dilemma. Either the market maker submits one large order which may perturb the

current price or the large order is split into smaller orders while running the risk that

others could front run the order if they see the pattern of buy and sell orders. Thus,

Hewlett’s model sought to combine order flow with the needs of brokers moving the

vast sums of capital tied up in FX markets.

One of the most interesting / pertinent works for this paper then came in 2012

from Filimonov and Sornette and their study of market endogeneity (whether price

changes are driven more by exogenous news events related to a firm or economy, or

endogenously by market movements caused by positive feedback mechanisms that

introduce correlation into the price changes). In particular this is when the branching

ratio of a Hawkes process becomes a major concern as Filimonov and Sornette use that

ratio as an appropriate proxy for market endogeneity. Bringing this framework to the E-

mini S&P 500 future market the two authors then expanded the power of high

frequency trading in setting prices. First, when considering E-mini data from 1998 to

2010 the dynamic branching ratio, estimated via Maximum Likelihood, Filimonov and

Sornette calculated increased from the low value of ~0.3 to as high as 0.9 and

consistently above 0.6 by 2004. However, while market endogeneity increased overall

during this time spikes did not occur during times of market stress that were

29

accompanied by fundamental exogenous news like the debt downgrades of Greece and

Portugal in 2010.

Contrasting with this fundamentally driven trading volume the authors do find a

hard to explain increase in the branching ratio during the Flash Crash of May 6th, 2010.

During a very short span of time (approximately 36 minutes total) the U.S. stock market

declined nearly 9% during seemingly mundane afternoon trading before rapidly

rebounding to close to market open prices. All of this occurred without any meaningful

exogeneous news and therefore quickly demonstrated a far above average branching

ratio for that trading day. Additionally, Filimonov and Sornette note that while no

evidence has been discovered to directly link high frequency trading to the start of the

Flash Crash it was associated with automated trading systems which might have

exacerbated the extreme market movements experienced that day. Finally, their findings

did reconfirm the increase of the model branching ratio alongside the rise of bigger and

more active high frequency trading shops.

Multivariate Hawkes processes are also of great interest and Fauth and Tudor

are one of the better examples of work in this arena. In particular the two researchers

confirm the intuitive expectation that price fluctuations also vary based on trading on

volume. Thus, when modeling stock trading activity with a marked, multivariate,

Hawkes process greater accuracy is found than when using price changes alone. Strong

evidence supports the conclusion that small quantities of shares do not affect the market

in the same way huge quantities do and that quantities and price changes are closely

linked. Additionally, Fauth and Tudor discuss the creation of a compound counting

30

process that allows for a growth process to change the underlying attributes of the

process over time.

Other Interesting Areas of Self-Exciting Point Process Research

One of the most interesting applications of self-exciting processes outside of

finance is the SEISMIC (Self-Exciting Model of Information Cascades) model.

SEISMIC is a self-exciting point process that Zhao, Erdogdu, He, Rajaraman, and

Leskovec from Stanford University created to model the final number of shares a social

post will receive based on its resharing history thus far. Not only does this paper create

a model very similar to the Hawkes process, but it also utilizes an infectiousness

parameter in order to determine when the underlying behavior is supercritical or

subcritical. Similar to a branching ratio the critical state determines whether SEISMIC

model can or cannot predict the eventual number of total shares. Thus, instead of seeing

changes in infectiousness as something to be measured like many finance paper the

SEISMIC authors use it as a barometer to determine the potential effectiveness of a

projection made at that time. Additionally, most tweets quickly fall to a subcritical state

allowing for SEISMIC to make a prediction for 98.20% of all tweets after observing

them for only 15 minutes.

SEISMIC therefore has many interesting properties in comparison to Hawkes

model, but like the following algorithms it also remains easy to implement. Three of

SEISMIC’s most notable attributes are that it is a generative model, has scalable

computation, and easy to interoperate. In order to implement SEISMIC only the time

history of reshares and the degrees of the resharing nodes are needed without any

parametric assumptions. Further, SEISMIC runs in linear computation time based on

31

the number of observed reshares and can be easily parallelized. Since the model

synthesizes all its past history into a single infectiousness parameter this parameter

conveys clear information about the information cascade and can be used as an input to

other applications.

32

Algorithms for Simulation and Estimation

Throughout this section effort will be made to focus upon the implementation

and intuition of the described processes. For a more rigorous treatment of the subject

matter references have been make to the necessary articles, and some results assumed in

order to maintain a more readable style. Overall, this style was deemed most consistent

with the tone of the thesis and in greatest conformity with its goal of reaching

undergraduate readers.

Poisson Processes

The first process simulation is a simple homogeneous Poisson process with rate

λ. Recall that a Poisson process with rate λ has a distribution of T(x), the interarrival

time function for the process, is T(x) = P(X ≤ x) = 1 − e−λx, x ≥ 0; E(X) = 1/λ. Since

this distribution function can be easily inverted with the natural log an algorithm for

simulating a Poisson process with rate λ up to time T is:

Homogeneous Poisson Process Algorithm (Sigman, 2016):

1. t = 0, N = 0

2. Generate U (U is a random variable in [0,1]).

3. t = t + [−(1/λ) ln (U)]. If t > T, then stop.

4. Set N = N + 1 and set tN = t.

5. Go back to 2.

Reviewing the algorithm it can be see that −1𝜆

ln(𝑈)is replicating the exponential decay

of the Poisson process. This is done by taking random values for U from [0,1], making

33

the natural log this number positive and scaling by the inverse of λ. An examples of the

values this takes on is:

Figure 6: Graph of −1𝜆

ln(𝑈) for λ = 5

Random values from this range are then added to the previous time in order to create the

Poisson process.

With only a few minor modifications this algorithm can then be used to simulate

two independent Poisson processes with rates λ1, λ2 up to time T:

Algorithm to Simulate Two Independent Poisson Processes (Sigman, 2016):

1. t = 0, t1 = 0, t2 = 0, N1 = 0, N2 = 0, set λ = λ1 + λ2, set p = λ1/λ.

2. Generate U.

3. t = t + [−(1/λ) ln (U)]. If t > T, then stop.

4. Generate U. If U ≤ p, then set N1 = N1 + 1 and set 𝑡𝑁1 = t; otherwise (U > p) set N2 = N2 + 1 and set 𝑡𝑁2 = t

5. Go back to 2.

00.10.20.30.40.50.60.70.80.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

34

Notice all that changed in this algorithm is that after the same first variable was created

a second was used to determine which Poisson process would be allocated the event.

Hence, the relative values of λ1 and λ2 became the determining factor in which process

would advance more rapidly than the other. Additionally this algorithm easily

generalizes to handle k > 2 independent Poisson processes. This is because all that

would need to happen is a further bracketing of the relative sizes of each λi so that

larger λ values may receive more points than smaller λ values.

However, as described in the point cluster section, the simulation of point

clusters requires a non-homogeneous Poisson process. In order to simulate a non-

homogenous Poisson process a thinning method may be used to slightly modify the

structure of the above algorithms. The algorithm to generate a non-homogenous Poisson

process with intensity λ(t) bounded by λ∗ up to time T with N(T) arrival times t1, ..., tN(T)

is then:

Non-homogenous Poisson Process With Intensity λ(t) That is Bounded by λ∗ Algorithm

(Sigman, 2016):

1. t = 0, N = 0

2. Generate U1

3. t = t + [−(1/λ∗) ln (U1)]. If t > T, then stop.

4. Generate a U2

5. If U2 ≤ λ(t)/λ∗, then set N = N + 1 and set tN = t.

6. Go back to 2.

Notice that this process is extraordinarily similar to the independent Poisson process

algorithm and simply checks to see if a point should be included or excluded from the

35

generated list. Now in order to prove this result we will begin with a nice fact about

partitioning Poisson processes.

Theorem (Partitioning a Poisson process): If X comes from a Poisson process with rate

α and if each object of X is, independently, type 1 or type 2 with probability p and q = 1

− p, then X1 comes from a Poisson process with rate pα, and X2 comes from a Poisson

process with rate qα and they are independent.

Proof (Sigman, 2016): We must show that

𝑃(𝑋1 = 𝑘,𝑋2 = 𝑙) = 𝑒−𝑝𝛼 (𝑝𝛼)𝑘

𝑘!𝑒−𝑞𝛼 (𝑞𝛼)𝑚

𝑚!. (1)

By the properties of Poisson processes we know that

𝑃(𝑋1 = 𝑘,𝑋2 = 𝑙) = 𝑃(𝑋1 = 𝑘,𝑋2 = 𝑘 + 𝑙)

= 𝑃(𝑋1 = 𝑘 | 𝑋 = 𝑘 + 𝑙) × 𝑃(𝑋 = 𝑘 + 𝑙).

Given X = k +m, it then follows that X1 ~ Binomial (k + m, p) and

𝑃(𝑋1 = 𝑘 | 𝑋 = 𝑘 + 𝑙) × 𝑃(𝑋 = 𝑘 + 𝑙) = (𝑘 + 𝑙)!𝑘!𝑙!

𝑒𝑘𝑞𝑚𝑒−𝛼𝛼𝑘+𝑚

(𝑘 + 𝑙)!

= 𝑒−𝛼 (𝑝𝛼)𝑘

𝑘!(𝑞𝛼)𝑚

𝑚!.

Since 1 = p + (1 - p) = p + q we then know that 𝑒𝛼 = 𝑒𝑝𝛼𝑒𝑞𝛼. This equality then shows

the above equation reduces to (1) and proves the result. ☐

This result then helps prove the initial result we sought:

Proof (Sigman, 2016): [Thinning works] Let {M(t)}be the counting process of the

λ∗ rate Poisson Process, and {N(t)}be the counting process of the thinned process. In

order to prove the result it must be shown that {N(t)} has independent increments and

that the increments are Poisson distributed with the correct mean, m(t).

36

First, it is true that {N(t)} has independent increments because {M(t)} has

independent increments and the thinning is done independently of {M(t)}. Thus,

{N(t)}inherits independence of increments from {M(t)}. So what is left to prove is that

for each t > 0, N(t) constructed from the thinning method has a Poisson distribution

with mean m(t) = ∫ 𝜆(𝑠)𝑡0 𝑑𝑠. Since M(t) is a Poisson distribution with mean λ*t it is

possible to partition M(t) into two types for each t > 0. If N(t) represents the accepted

points and R(t) the rejected ones, M(t) will be fully described by these two functions.

Hence since M(t) can be partitioned, it follows from the above Theorem that N(t) has

the desired Poisson distribution. Specifically the conditional on M(t) = n, it is possible

to take n unordered arrival times as i.i.d. uniform (0, t) random variables. Thus an

arrival, denoted by V ~ Uniform(0, t) will be accepted with conditional probability

λ(v)/λ*, conditional on V = v. Therefore the unconditional probability of acceptance is:

𝑒 = 𝑒(𝑡) = 𝐸 �λ(V)λ∗

� = 1𝜆∗𝑡

� 𝜆(𝑠)𝑡

0 𝑑𝑠

and we conclude from the partitioning theorem that N(t) has a Poisson distribution with

mean λ*tp = m(t). ☐

Univariate Hawkes Process with Exponentially Decaying Intensity

For this first Hawkes process method we will consider a Hawkes process with

exponentially decaying intensity. While this might be a special case of the more general

Hawkes process it is numerically efficient to calculate and the “most widely

implemented in practice” (Dassios, Angelos, and Zhao, 2013). The algorithm can be

implemented quite easily because the random interarrival-times between events in the

process are simulated by decomposing each event into two independent random

37

variables without inverting the underlying cumulative distribution function. Later this

will be discussed as one of the frequent numerical limiting factors to implementing

Hawkes processes, and therefore avoiding the inversion of the cumulative distribution

function makes this implementation of Hawkes processes simpler and faster.

Additionally, this algorithm does not require stationarity. A few necessary definitions

are:

● a > 0 is the constant reversion level (i.e. the background intensity)

● λ0 > 0 is the initial intensity at time t = 0

● δ > 0 is the constant rate of exponential decay

● {Yk}k = 1,2,.... are sizes of self-excited intensity jumps, a sequence of i.i.d positive

random variables with distribution function G(y), y > 0

Written in our previous form this would therefore be the Hawkes process intensity

𝜆𝑡(𝑡) = 𝑎 + (𝜆0 − 𝑎)𝑒−𝛿𝑡 + � 𝑌𝑖𝑒−𝛿(𝑡 − 𝑡𝑖)

𝑡𝑖 < 𝑡.

Univariate Hawkes Algorithm4: The simulation algorithm for one sample path of one-

dimensional Hawkes process with exponentially decaying intensity {(Nt, λt)}t > 0

conditional on λ0 and N0 = 0, with intensity jump-size distribution Y ~ G and K event-

times {T1, T2, ..., TK}:

1. Set the initial conditions: T0 = 0, 𝜆𝑇0± = 𝜆0 > 𝑎,𝑑0 = 0 and 𝑘 ∈ {0,1, … ,𝐾 − 1}

2. Simulate the (k + 1)th interarrival-time Sk+1 by

Sk + 1 = �𝑆𝑘+1

(1) ∧ 𝑆𝑘+1(2) 𝐷𝑘+1 > 0

𝑆𝑘+1(2) 𝐷𝑘+1 < 0

where

38

𝐷𝑘+1 = 1 +𝛿 ln𝑈1𝜆𝑇+𝑘 − 𝑎

, 𝑈1~ U[0,1]

and

𝑆(1)𝑘+1 = −

1𝛿

ln𝐷𝑘+1 ,

𝑆(2)𝑘+1 = − 1

𝑚ln𝑈2 , 𝑈2 ~ U[0,1].

Commentary: This then means that:

𝑆𝑘+1(1) ∧ 𝑆𝑘+1

(2) = −1𝛿

ln�1 +𝛿 ln𝑈1𝜆𝑇+𝑘 − 𝑎

� × −1𝑎

ln𝑈2

= 1𝛿𝑎

ln�1 +𝛿 ln𝑈1𝜆𝑇+𝑘 − 𝑎

� × ln𝑈2

where 𝛿 is the constant rate of exponential decay and 𝜆𝑇+𝑘 − 𝑎 is the current

gap between the constant background intensity and the self-exciting endogenous

excitement.

Figure 7: Example of a Generic Exponential Intensity Decay

0

1

2

3

4

5

6

7

0 0.5 1

Background / Exogenous Intensity

EndogenousIntensity

More likely to get a endogenous "child" event

More likely to get an exogenous "immigrant" event

39

Figure 7 helps demonstrate the changes in intensity by showing the large amount

of endogenous intensity on the left, and the more dominant exogenous intensity

on the right. Hence when the quantity 1 + 𝛿 ln𝑈1𝜆𝑇+𝑘− 𝑚

is large we are more likely to

see a new child (endogenous event), and when this value is small a new

immigrant (exogenous event) is more likely. If 1 + 𝛿 ln𝑈1𝜆𝑇+𝑘− 𝑚

is positive

then 𝐷𝑘+1 > 0, we can solve the first natural log and multiply it with second

logarithm to find the random value of the exogenous and endogenous points

together. The random value of these two processes are then scaled by the

exponential decay factor and the background intensity through 1𝛿𝑚

. However, if

this is negative then an exogenous point should be generated as the ratio was

quite small (i.e. the endogenous intensity was small in relation to the constant

exogenous intensity).

3. Record the (k + 1)th event-time Tk+1 in the intensity process λt by

Tk+1 = Tk + Sk+1

4. Record the change at the event-time Tk+1 in the intensity process λt by

𝜆𝑇+𝑘+1 = 𝜆𝑇−𝑘+1 + 𝑌𝑘+1, Yk+1 ~ G

where

𝜆𝑇−𝑘+1 = �𝜆𝑇+𝑘 − 𝑎�𝑒−𝛿(𝑇𝑘+1−𝑇𝑘) + 𝑎.

Commentary: Recall that {Yk}k = 1,2,... are sizes of self-excited intensity jumps, a

sequence of i.i.d. positive random variables with distribution function G(y). So

𝑌𝑘+1 is the value by which the endogenous intensity of the process is increasing.

However, this is an exponentially time decaying process with faster decay

40

happening the larger the intensity currently is. This is why the calculation of

𝜆𝑇−𝑘+1 not only has an exponential decay value, but also multiplies this by the

size of the exogenous intensity minus the background exogenous intensity.

5. Record the change at the event-time Tk+1 in the point process Nt by

𝑑𝑇+𝑘+1 = 𝑑𝑇−𝑘+1 + 1.

For the proof of this result in its entirety the reader is encouraged to consult

Dassios and Zhao (2013). Here we will give a rough overview of the proof, but will

simply assume that the inverse of the cumulative distribution function can be replaced

with two independent variables S(1)k+1 and S(2)

k+1. While avoiding this proof loses some

of the rigor of the result it makes the proof far more readable and focuses attention on

the benefits this algorithm brings to implementation. The importance of this result for

implementation that it allows for exact simulation,1 which avoids “introducing

discretization bias for associated estimators,” without numerical evaluation of the

inverse of analytic distribution functions. Inverting analytic distribution functions

generally requires using Brent’s method and involves intensive computations.

Algorithmically solving this problem therefore keeps the benefits of the most precise

methods of Hawkes process simulation while remaining easily computable.

Proof: Given a kth event-time Tk, the point process then has the intensity process

{λt}𝑇𝑘≤ 𝑡 ≤𝑇𝑘+𝑆𝑘+1following the ODE

𝑑𝜆𝑡𝑑𝑡

= −𝛿(𝜆𝑡 − 𝑎).

With the initial condition λt|𝑡=𝑇𝑘 = 𝜆𝑇𝑘 .The above ODE has a unique solution given by

1 Here ‘exact’ simulation means a method of drawing an unbiased associated estimator thought the entire

simulation process.

41

𝜆𝑡 = (𝜆𝑇+𝑘 − 𝑎) 𝑒−𝛿(𝑡−𝑇𝑘) + 𝑎, 𝛿𝑘 ≤ 𝑡 ≤ 𝛿𝑘 + 𝑆𝑘+1

and the cumulative distribution function of the (k+1)th interarrival-time Sk+1 is given by

𝐹𝑆𝑘+1(𝑠) = 𝑃{𝑆𝑘+1 ≤ 𝑠}

= 1 - P{Sk+1 > s}

= 1 − 𝑃{𝑑𝑇𝑘+𝑠 − 𝑑𝑇𝑘 = 0}

= 1 − exp �−∫ 𝜆𝑐𝑇𝑘+2𝑇𝑘

𝑑𝑑�

= 1 − exp �−∫ 𝜆𝑇𝑘+𝑣+𝑠0 𝑑𝑑�

= 1 − exp �−�𝜆𝑇+𝑘 − 𝑎� 1−𝑒−𝛿𝛿

𝛿− 𝑎𝑠�.

By the inverse transformation method it follows that

Sk+1 𝐹−1𝑆𝑘+1(𝑈), U ~ [0,1].

However, inverting the function 𝐹𝑆𝑘+1(⋅)can be avoided by decomposing Sk+1 into two

simpler and independent random variables S(1)k+1 and S(2)

k+1 via

Sk+1 S(1)k+1 ∧S(2)

k+1.

Assuming the ability to do this then means that the next time interval will be

determined by the two random variables S(1)k+1 and S(2)

k+1. These together then

represent 𝐹−1𝑆𝑘+1(𝑈) and give the next interarrival length. Therefore the (k+1)th event-

time Tk+1 in the Hawkes process will be given by

Tk+1 = Tk + Sk+1

and the change in λt and Nt at time Tk+1 then can be derived as 𝜆𝑇+𝑘+1 = 𝜆𝑇−𝑘+1 + 𝑌𝑘+1,

Yk+1 ~ G and 𝑑𝑇+𝑘+1 = 𝑑𝑇−𝑘+1 + 1 from steps 4 and 5 of the algorithm respectively. ☐

42

Multivariate Hawkes Process with Exponentially Decaying Intensity

Making only a couple modifications to the univariate Hawkes process

algorithm it is possible to extend the algorithm to multi-dimensional cases. With K joint

event-times {T1, T2, ..., TK} before time t a D-dimensional point process {N[j]t}j=1,2,...,D

where N[j]t = {T[j]

k}k=1,2,.... can be defined by the following underlying intensity process:

𝜆[𝑚]𝑡 = 𝑎[𝑚] + �𝜆[𝑚]

0 − 𝑎[𝑚]� 𝑒−𝛿[𝑗]𝑡 + � � 𝑌[𝑚,𝑙]𝑘𝑒−𝛿

[𝑗](𝑡−𝑇[𝑙]𝑘)

0≤ 𝑇[𝑙]𝑘< 𝑡

𝐷

𝑙=1

with 𝑗 ∈ {1,2, . . . ,𝐷}.

Where {Y[j,l]k}j=l are the sizes of self-excited intensity jumps and {Y[j,l]

k}j≠l are sizes of

cross-excited jumps, and they are measurements of the impacts of self-contagion and

cross-contagion respectively. Upon the arrival of an event in point process N[l]t, note

that each marginal intensity process {λ[j]t}j=1,2,....,D experiences a simultaneous intensity

jump of positive random size, and these intensity jumps can be either dependent or

independent.

Multivariate Hawkes Process Algorithm (Dassios, Angelos, and Zhao, 2013): The

simulation algorithm for one sample path of a D-dimensional Hawkes process with

exponentially decaying intensity {(N[j]t, λ[j]

t)}t > 0 for j ∊ {1,2,...,D} conditional on λ[j]0

and N[j]0 = 0, with K joint event-times {T1, T2, ..., TK} in intensity processes:

1. Set the initial conditions T0 = 0, 𝜆[𝑚]𝑇±

0= 𝜆[𝑚]

0 > 𝑎[𝑚], N[j]0, j ∊ {1,2,...,D}

and k ∊ {0,1,2,...,K-1}

2. Simulate the (k+1)th interarrival-time Wk+1 by

Wk+1 = min{S[1]k+1,S[2]

k+1,....,S[D]k+1}

where

43

Wk+1 = Sll]k+1

and each S[j]k+1 can be simulated in the same way as Sk+1 as given by Step 2 of

the univariate algorithm

Commentary: Notice that all that has changed in this situation is that

interarrival time happens much faster because the algorithm merely picks the

process that takes the least amount of time to cause an event. For actual

implementations this means that the decay rate for time will generally be faster

here than in the univariate case. This will then correctly allocate effects to other

variables, and ensure the process continues at a reasonable rate.

3. Record the (k+1)th event-time Tk+1 in the intensity process λ[j]t by

Tk+1 = Tk + Wk+1

4. Record the change at the event-time Tk+1 in the intensity process N[j]t by

𝑑[𝑚]𝑇+𝑘+1 = �

𝑑[𝑚]𝑇−𝑘+1 + 1 𝑗 = 𝑙

𝑑[𝑚]

𝑇−𝑘+1 𝑗 ≠ 𝑙 𝑗 ∊ {1,2, . . . ,𝐷}

Commentary: This is simply a fancy way of updating a list of values to ensure

that that we allocate the event to the right sub-process. Once the match has been

made then the algorithm just pulls the right values in order to increase the right

intensity.

5. Record the change at the event-time Tk+1 in the intensity process λ[j]t by

𝜆[𝑚]𝑇+𝑘+1 = 𝜆[𝑚]

𝑇−𝑘+1 + 𝑌[𝑚,𝑙]𝑘+1, j ∊ {1,2,...,D}

where

𝜆[𝑚]𝑇−𝑘+1 = �𝜆[𝑚]

𝑇+𝑘 − 𝑎[𝑚]�𝑒−𝛿[𝑗](𝑇𝑘+1−𝑇𝑘) + 𝑎[𝑚].

44

Estimation of a Power Kernel

When seeking to estimate a point cluster with a non-homogenous Poisson

process it is important to be able to find an estimate for λ(t) in order to estimate the

limiting factor for point acceptance appropriately. Previous work has been done with

the exponential kernel, but in this situation the limiting factor can be estimated more

easily with a power kernel. In particular the following method utilizes easy to follow

steps that can be used on any list formatted data. Estimating λ(t) through this method is

done by using the power law function

M(t) = atb

which has the intensity function

𝜆(𝑡) =𝑑𝑑(𝑡)𝑑𝑡

= 𝑎𝑎𝑡𝑏−1.

Note that if 0 < b < 1, the intensity function is decreasing and if b > 1 the rate is

increasing. Then the modified maximum likelihood estimation2 (MLE) coefficients,

conditioned on time tn, for the power model are (Tobias and Trindade, 2012)

𝑎� = 𝑛−2∑ ln𝑡𝑛𝑡𝑛−1𝑖=1

, 𝑎� = 𝑛𝑡𝑏�𝑛

.

Similarly considering a fixed time T, so that the number of events N by time T is

random, we can condition on N to get the MLEs as

𝑎� = 𝑁−1∑ 𝑙𝑛𝑇𝑡𝑖𝑁𝑖=1

, 𝑎� = 𝑁𝑇𝑏�

.

These estimations can then be improved by estimating the kernel on k different

time intervals, with similar conditions, to improve the estimates of the model

2 The values that maximize the probability of obtaining a particular set of data, given the chosen

probability distribution model.

45

parameters. When performing these tests it’s possible to limit both by time and number

of events to occur. Let Tq denote the length of time considered for the qth time interval,

q = 1,2,....,k and let nq denote the total number of events in the qth sample by the time

Tq. Let tiq be the ith event time within the qth sample. Now we will introduce a new

variable Nq, which equal nq if the data from the qth sample are time limited or equals nq

- 1 if the data on the qth system are event limited. Crow (1974) demonstrates that

conditioning on either the number of events in each window or the length of time of

each the unbiased MLE for b can be expressed in closed form as

𝑎� =𝑑𝑠 − 1

∑ ∑ 𝑙𝑛𝛿𝑞𝑡𝑖𝑞

𝑁𝑞𝑖=1

𝑘𝑞=1

where

𝑑𝑠 = � 𝑑𝑞𝑘

𝑞=1.

The modified MLE for a is then

𝑎� =∑ 𝑛𝑞𝑘𝑞=1

∑ 𝛿𝑞𝑏�𝑘

𝑞=1

.

Hence this process provides a way of using given data to estimate values for a and b

that may be used to generate λ(t) for a non-homogeneous Poisson process that produces

children for point clusters.

46

Implemented Processes

In an Excel workbook3 each of these algorithms, except the multivariate

Hawkes process were simulated. Some of the simulated results are:

Figure 8: Homogenous Poisson Process with λ = 4 and T = 50

Figure 9: Non-homogenous Poisson Process: λ = 7.5, T = 7.5, & λ(t) = 𝑛

𝜏𝑒𝑒𝑒 �− 𝑡

𝜏�

3 https://drive.google.com/open?id=0BxM3BpLkURvVVERBeHdQUnotYk0

0

5

10

15

20

25

30

35

40

1 15 30 45 60 75 90 105 120 135 150

0

1

2

3

4

5

6

7

0 2 4 6 8 10 12 14

47

Figure 10: Intensity of Univariate Hawkes Process: a = 0.45, δ = 0.25, λ0 = 0.9, &

Intensity Jumps of Size 0.40

Figure 11: Histogram of Number of Events per Unit Time for Univariate Hawkes

Process: a = 0.45, δ = 0.25, λ0 = 0.9, & Intensity Jumps of Size 0.40

Future steps for this work would then be to use financial data to better estimate

parameters and kernels for these models. In some other fields it might be possible that

-123456789

10111213

- 20 40 60 80 100 120 140 160 180 200

0

5

10

15

20

25

3.7 29.4 55.1 80.8 106.5 132.3 158.0 183.7

48

there are generally accepted values for how things are modeled, but financial models are

almost always built around empirical data. This is also the case in many other fields

where the relevant data is available to help tune and test a model before using for

simulations or forecasts. Better tuning then ensures researchers find the best relationship

between intensity (Figure 9) and the actual number of events occurring (Figure 10).

While lacking such data does not immediately preclude a Hawkes process from being

used to model a point process it does make the results of the model far less certain.

Overall this reinforces that Hawkes processes are very robust and flexible models, but

processes that are frequently most useful with adequate empirical data.

49

Future Work

Improve Simulation Implementations

Creating results in R, Matlab, or with a coding language would be one of the

first necessary tasks. Accumulating the knowledge to do this would take a great deal of

time, but would be necessary to continue research in an applied setting. Next, estimation

methods would be researched to better estimate the parameters necessary for modeling

empirical data. Then within one particular financial market a great deal of testing would

need to be done to determine what time horizon to study, what kernel to use, how to

best tune the model, what additional attributes would be meaningful for a multivariate

Hawkes process, and how accurate models are for that arena. Having completed this

survey theoretical modifications to the kernel would most likely be the most fruitful

area of study, but finding numerically simple solutions for currently complex ones

would also be an interesting pursuit.

Research such as this would then provide even greater clarity on the

characteristics of some of the most important financial markets in the world. Most

importantly work in this arena can explore the often dramatic reactions of financial

markets to external events. Additionally, further research should help give a better

image of the impact high frequency trading is having on global financial markets. While

enormous development have occurred in high frequency trading some feel laws and

regulators have failed to keep pace with these progressions. Studying the effects of

highly computerized trading through Hawkes processes might therefore be quite helpful

in modernizing outsiders understanding of this arena.

50

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